> Isn't it an oversimplification to say that the motivation for the
> axioms of set theory is only philosophical (hence not mathematical)?
>> Isn't the need for a foundational framework at least as much
> mathematical as philosophical? Does the fact that one has to face
> problems and methods that lay outside the deductive paradigm change
> the mathematical character of foundational research?
The last question is not very clear to me. But, anyway...
>> I think that the deductive paradigm is too restrictive, assigning the
> mathematician a unique role as a "theorem prover", and so explicitly
> excluding the role as a "theory framer" every other kind of scientist
> must play. Why shouldn't mathematicians explore different theories
> through their consequences and choose those that best meet their
> needs, just like any other kind of scientists do?
I essentially agree with your, as I consider, rhetorical questions and
the above opinion except I would not consider the deductive paradigm so
restrictive (just as deducing theorems). I would preset things slightly
differently.
My understanding (already presented in the FOM list) is that
mathematics is devoted to formalisation of our thought and intuition
(because formalisation can provide an extraordinary, miraculous
mechanising/automating/powering/etc tools for our thought).
I consider this as a (renewed) formalist view on mathematics.
The best historical example is the Newton-Leibniz Calculus.
Deriving theorems in these formalisms is only a particular case
of this activity exploring (as you write) a given formalism.
Compare, what is scientifically greater: creating the Calculus
or proving any particular theorem, say, on differential equations,
both kind of activities being genuine mathematics. In fact,
creating an interesting formalism is hardly possible without
exploring it, at least to demonstrate its potential.
I believe this view puts things in the right way.
Mathematics is, indeed, more than just deducing theorems.
Vladimir Sazonov